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Infinitely many positive solutions for the Neumann problem involving the p-Laplacian

100%

Anello G. , Cordaro G.

nr 2

221-231

We present two results on existence of infinitely many positive solutions to the Neumann problem ⎧ $-Δ_{p}u + λ(x)|u|^{p-2}u = μf(x,u)$ in Ω, ⎨ ⎩ ∂u/∂ν = 0 on ∂Ω, where $Ω ⊂ ℝ^{N}$ is a bounded open set with sufficiently smooth boundary ∂Ω, ν is the outer unit normal vector to ∂Ω, p > 1, μ > 0, $λ ∈ L^{∞}(Ω)$ with $essinf_{x∈Ω} λ(x) > 0$ and f: Ω × ℝ → ℝ is a Carathéodory function. Our results ensure the existence of a sequence of nonzero and nonnegative weak solutions to the above problem.

An existence and localization theorem for the solutions of a Dirichlet problem

100%

Anello G. , Cordaro G.

nr 2

107-112

We establish an existence theorem for a Dirichlet problem with homogeneous boundary conditions by using a general variational principle of Ricceri.

Parametrization of Riemann-measurable selections for multifunctions of two variables with application to differential inclusions

100%

Anello G. , Cubiotti P.

nr 2

179-187

We consider a multifunction $F:T×X → 2^E$, where T, X and E are separable metric spaces, with E complete. Assuming that F is jointly measurable in the product and a.e. lower semicontinuous in the second variable, we establish the existence of a selection for F which is measurable with respect to the first variable and a.e. continuous with respect to the second one. Our result is in the spirit of [11], where multifunctions of only one variable are considered.